Calculus of Variations and Geometric Measure Theory

L. Rizzi - G. Stefani

Failure of curvature-dimension conditions on sub-Riemannian manifolds via tangent isometries

created by stefani on 02 Jan 2023
modified on 07 Aug 2023


Published Paper

Inserted: 2 jan 2023
Last Updated: 7 aug 2023

Journal: J. Funct. Anal.
Volume: 285
Number: 9
Pages: Paper No. 110099
Year: 2023

ArXiv: 2301.00735 PDF


We prove that, on any sub-Riemannian manifold endowed with a positive smooth measure, the Bakry-Émery inequality for the corresponding sub-Laplacian, \[ \frac{1}{2}\Delta(\Vert\nabla u\Vert^2) \geq \mathrm{g}(\nabla u,\nabla \Delta u) + K \Vert\nabla u \Vert^2, \quad K\in\mathbb R, \] implies the existence of enough Killing vector fields on the tangent cone to force the latter to be Euclidean at each point, yielding the failure of the curvature-dimension condition in full generality. Our approach does not apply to non-strictly-positive measures. In fact, we prove that the weighted Grushin plane does not satisfy any curvature-dimension condition, but, nevertheless, does admit an a.e. pointwise version of the Bakry-Émery inequality. As recently observed by Pan and Montgomery, one half of the weighted Grushin plane satisfies the $\mathsf{RCD}(0,N)$ condition, yielding a counterexample to gluing theorems in the $\mathsf{RCD}$ setting.

Keywords: Grushin plane, Sub-Riemannian manifold, $\mathsf{CD}(K,\infty)$ condition, Bakry-Émery inequality, infinitesimally Hilbertian, privileged coordinates